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editing

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allocated for Michael WallnerNumber of nondeterministic Dyck excursions of length 2*n.

1, 4, 28, 224, 1888, 16320, 143040, 1264128, 11230720, 100124672, 894785536, 8010072064, 71794294784, 644079468544, 5782109208576, 51934915067904, 466666751655936, 4194593964294144, 37711993926844416, 339119962067042304, 3049961818869989376, 27434013235435536384, 246790115075341418496, 2220247697892210376704, 19975783025562120880128, 179733771152022967418880, 1617241407368395227136000, 14552443310731262047551488, 130951393636087635116032000, 1178406784261054933497806848, 10604480765474749257491677184, 95431366209911167132364701696, 858814148177068828685160677376, 7728808208374734047693343031296, 69555313290691071405707841503232, 625967559359573879445260169379840, 5633476521253927059293232819077120, 50699515179992463538944962631565312, 456282034154331246112587938277621760, 4106433859885259820954288017270374400, 36957101861729430088935946411929763840, 332607738771623823169979457927810908160, 2993422064100863827711139014397059399680, 26940431722213167698859429996718624604160, 242461054734615444434497895555246755676160, 2182127631103410439262156382584020528005120, 19638979711261702038665452055520637700014080, 176749510428700665887831323575551282852659200, 1590735476979942008520977543307429762752839680, 14316540925923227827759637124731179361686781952

0,2

In nondeterministic walks (N-walks) the steps are sets and called N-steps. N-walks start at 0 and are concatenations of such N-steps such that all possible extensions are explored in parallel. The nondeterministic Dyck step set is { {-1}, {1}, {-1,1} }. Such an N-walk is called an N-excursion if it contains at least one trajectory that is a classical excursion, i.e., never crosses the x-axis, and starts and ends at 0 (for more details see the de Panafieu-Wallner article).

Élie de Panafieu and Michael Wallner, <a href="https://arxiv.org/abs/2311.03234">Combinatorics of nondeterministic walks</a>, arXiv:2311.03234 [math.CO], 2023.

G.f.: (1-8*x-(1-12*x)*sqrt(1-8*x))/(8*x*(1-9*x)).

The a(1)=4 N-bridges of length 2 are

/ /

/\, /\, /\, /\

\ \/

\ \

Cf. A151281 (Nondeterministic Dyck meanders), A368164 (Nondeterministic Dyck bridges), A000244 (Nondeterministic Dyck walks).

allocated

nonn

Michael Wallner, Dec 18 2023

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editing

allocated for Michael Wallner

allocated

approved